THIS WORK IS YET TO BE REVIEWED
This paper contains estimates for the reproductive number \(R_{t,m}\) over time \(t\) in various countries \(m\). It also shows the same for provinces within South Africa.. This is done using the methodology as described in [1]. These have been implemented in R using EpiEstim package [2] which is what is used here.
This paper and it’s results should be updated roughly daily and is available online.
As this paper is updated over time this section will summarise significant changes. The code producing this paper is tracked using Git. The Git commit hash for this project at the time of generating this paper was 8f47d1372fda5d4e2e1a09f873a84506cdc142f1.
2020-06-12
2020-06-13
2020-06-14
2020-06-15
2020-06-17
All countries available in the data will be analysed but also the provinces of South Africa. This paper produces charts for the following countries only:
Data is downloaded from the Git repository associated with [3].
In the case data file row 21 and 32 contain no provincial details. We estimated it by spreading the national total to the provinces in proportion to the combined mixture of the prior day and the next day.
Further fixes are applied to both case and death data:
SA column is added as the sum of the new per province data.Data for other countries are downloaded from [4]. Minor formatting is applied to get the data ready for further processing.
For Spain death data reporting of deaths appear to have significantly changed and we therefore exclude the data from this analysis [5].
Google released mobility data indexes that deviations from a baseline of movement during 3 January to 6 February 2020 [6]. This data is downloaded and prepared for linking to our estimates of \(R_{t,m}\). The data is made available as a comma-separated file which is updated regularly but not daily. Data is sometimes up to a week behind. For countries other than South Africa only the country level mobility data is kept. I.e. we ignore regional data within countries. For South Africa provincial data is also kept.
We also calculate an average mobility index per [7]. This index averages the mobility indexes but exclude the following indexes:
Below we plot cumulative case count on a log scale by province:
Below we plot the cumulative deaths by province on a log scale:
Below we plot average mobility indexes by province:
Below we plot cumulative case count on a log scale by country:
Below we plot the cumulative deaths by country on a log scale:
Below we plot average mobility indexes by country:
This analysis follows the follows the method proposed [1].
Essentially this models the following relationship:
\(E(I_{t,m})=R_{t,m}\sum_{s=1}^tI_{t-s,m}w_{s}\) where:
The formulation of instantaneous rate of transmission proposed in [1] is convenient because it ensures that \(R_{t,m}\) only depends on information that is known at time \(t\). We do not need to know about future transmission.
To do further analysis an serial interval assumption is needed (\(w_{s}\)). The serial interval is taken from [8]. It’s assumed to be Gamma distributed with mean of 6.48 and standard deviation of 3.83. This corresponds to the effective infectiousness of an individual since acquiring the infection themselves.
We plot this serial distribution below:
[1] describes a choice of a time window to estimate \(R_{t,m}\). Based on this analysis we chose a time window of 7-days (\(\tau\) using notation of [1]). During this window \(R_{t,m}\) is assumed to be constant. However information prior that is taken into account.
Note that the time window size does not interact with the serial interval as information prior to the window is taken into account. It’s just the period during which \(R_{t,m}\) is assumed to be constant.
Based on the table in Appendix 2 of [1] we believe a window of 7-days to be reasonable as long as that window contains 12 or more cases or deaths.
The analysis works backwards and fits \(R_{t,m}\) values for whole weeks (ending on the last date in the data) using the EpiEstim package.
So \(t\) is based on the date of reporting (be that cases or deaths) and \(m\) is the country or province. Two values are estimated for each country:
Note that the time periods are left unadjusted, though in reality the \(R_{t,m}^{deaths}\) should be shifted back approximately 2 weeks relative to \(R_{t,m}^{cases}\).
Linking the mobility data should be done in a way that corresponds to the time the infection should have occurred not the time of death. For cases we assume 6 days from infection to case being diagnosed plus a further 7-day reporting delay. For death we assume 23 days from infection to death and a further 7-day reporting delay as well.
Mobility data is then averaged over the weeks corresponding to the \(R_{t,m}\) shifted as above.
TO DO
Below current (last weekly) \(R_{t,m}\) estimates are tabulated.
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| EC | cases | 9,300 | 2020-06-29 | 1.4 | 1.4 | 1.4 |
| EC | deaths | 94 | 2020-06-29 | 0.9 | 1.1 | 1.4 |
| FS | cases | 605 | 2020-06-29 | 1.8 | 2.0 | 2.1 |
| GP | cases | 17,499 | 2020-06-29 | 1.5 | 1.6 | 1.6 |
| GP | deaths | 58 | 2020-06-29 | 0.9 | 1.2 | 1.5 |
| KZN | cases | 3,763 | 2020-06-29 | 2.1 | 2.1 | 2.2 |
| KZN | deaths | 28 | 2020-06-29 | 1.0 | 1.4 | 2.0 |
| LP | cases | 429 | 2020-06-29 | 1.5 | 1.7 | 1.8 |
| MP | cases | 493 | 2020-06-29 | 1.4 | 1.6 | 1.7 |
| NC | cases | 150 | 2020-06-29 | 1.5 | 1.7 | 2.0 |
| NW | cases | 1,616 | 2020-06-29 | 1.3 | 1.4 | 1.4 |
| SA | cases | 42,674 | 2020-06-29 | 1.4 | 1.4 | 1.4 |
| SA | deaths | 538 | 2020-06-29 | 1.0 | 1.1 | 1.2 |
| WC | cases | 8,819 | 2020-06-29 | 0.9 | 0.9 | 1.0 |
| WC | deaths | 349 | 2020-06-29 | 0.9 | 1.0 | 1.1 |
The table below contains \(R_{t,m}\) estimates for the last week available. These estimates are based either on cases or deaths and a 95% confidence interval is shown.
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Brazil | cases | 259,105 | 2020-06-29 | 1.1 | 1.1 | 1.1 |
| Brazil | deaths | 7,005 | 2020-06-29 | 0.9 | 0.9 | 1.0 |
| Canada | cases | 1,913 | 2020-06-29 | 0.7 | 0.8 | 0.8 |
| Canada | deaths | 92 | 2020-06-29 | 0.3 | 0.4 | 0.4 |
| Chile | cases | 29,627 | 2020-06-29 | 0.6 | 0.6 | 0.6 |
| Chile | deaths | 1,030 | 2020-06-29 | 0.9 | 0.9 | 1.0 |
| India | cases | 123,036 | 2020-06-29 | 1.3 | 1.3 | 1.3 |
| India | deaths | 2,776 | 2020-06-29 | 0.8 | 0.8 | 0.9 |
| Ireland | cases | 60 | 2020-06-29 | 0.6 | 0.7 | 0.9 |
| Ireland | deaths | 20 | 2020-06-29 | 0.8 | 1.3 | 1.9 |
| Italy | cases | 1,811 | 2020-06-29 | 1.0 | 1.0 | 1.1 |
| Italy | deaths | 135 | 2020-06-29 | 0.4 | 0.5 | 0.6 |
| Peru | cases | 24,483 | 2020-06-29 | 0.9 | 0.9 | 1.0 |
| Peru | deaths | 1,272 | 2020-06-29 | 0.9 | 1.0 | 1.0 |
| South_Africa | cases | 40,832 | 2020-06-29 | 1.4 | 1.4 | 1.4 |
| South_Africa | deaths | 526 | 2020-06-29 | 1.0 | 1.1 | 1.1 |
| Spain | cases | 2,498 | 2020-06-28 | 1.0 | 1.0 | 1.1 |
| United_Kingdom | cases | 6,820 | 2020-06-29 | 0.8 | 0.8 | 0.9 |
| United_Kingdom | deaths | 918 | 2020-06-29 | 0.8 | 0.8 | 0.9 |
Below we show various extremes of \(R_{t,m}\) where counts (deaths or cases) exceed 50 in the last week.
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Canada | deaths | 92 | 2020-06-29 | 0.3 | 0.4 | 0.4 |
| Italy | deaths | 135 | 2020-06-29 | 0.4 | 0.5 | 0.6 |
| France | deaths | 138 | 2020-06-29 | 0.5 | 0.6 | 0.7 |
| Sudan | deaths | 51 | 2020-06-29 | 0.5 | 0.6 | 0.8 |
| Ecuador | deaths | 206 | 2020-06-29 | 0.6 | 0.7 | 0.8 |
| Pakistan | deaths | 577 | 2020-06-29 | 0.7 | 0.8 | 0.8 |
| Germany | deaths | 76 | 2020-06-29 | 0.6 | 0.8 | 0.9 |
| Algeria | deaths | 52 | 2020-06-29 | 0.6 | 0.8 | 1.0 |
| Poland | deaths | 82 | 2020-06-29 | 0.6 | 0.8 | 1.0 |
| India | deaths | 2,776 | 2020-06-29 | 0.8 | 0.8 | 0.9 |
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Nicaragua | cases | 156 | 2020-06-29 | 0.4 | 0.4 | 0.5 |
| Sudan | cases | 678 | 2020-06-29 | 0.4 | 0.5 | 0.5 |
| Djibouti | cases | 61 | 2020-06-29 | 0.4 | 0.5 | 0.7 |
| Chile | cases | 29,627 | 2020-06-29 | 0.6 | 0.6 | 0.6 |
| Malaysia | cases | 62 | 2020-06-29 | 0.5 | 0.6 | 0.7 |
| Afghanistan | cases | 2,134 | 2020-06-29 | 0.6 | 0.6 | 0.6 |
| Cameroon | cases | 982 | 2020-06-29 | 0.6 | 0.6 | 0.6 |
| Zimbabwe | cases | 78 | 2020-06-29 | 0.5 | 0.6 | 0.8 |
| Somalia | cases | 115 | 2020-06-29 | 0.5 | 0.6 | 0.8 |
| South_Sudan | cases | 107 | 2020-06-29 | 0.5 | 0.7 | 0.8 |
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| El_Salvador | deaths | 57 | 2020-06-29 | 1.4 | 1.8 | 2.3 |
| Honduras | deaths | 116 | 2020-06-29 | 1.3 | 1.6 | 1.8 |
| Colombia | deaths | 869 | 2020-06-29 | 1.4 | 1.5 | 1.6 |
| Kazakhstan | deaths | 63 | 2020-06-29 | 1.1 | 1.5 | 1.8 |
| Panama | deaths | 103 | 2020-06-29 | 1.1 | 1.4 | 1.6 |
| Iraq | deaths | 656 | 2020-06-29 | 1.3 | 1.4 | 1.5 |
| Bolivia | deaths | 241 | 2020-06-29 | 1.2 | 1.3 | 1.5 |
| Guatemala | deaths | 196 | 2020-06-29 | 1.1 | 1.2 | 1.4 |
| Iran | deaths | 885 | 2020-06-29 | 1.1 | 1.2 | 1.2 |
| Mexico | deaths | 4,823 | 2020-06-29 | 1.1 | 1.1 | 1.1 |
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Western_Sahara | cases | 183 | 2020-06-29 | 31.5 | 36.6 | 42.1 |
| Namibia | cases | 128 | 2020-06-29 | 3.5 | 4.2 | 4.9 |
| Paraguay | cases | 748 | 2020-06-29 | 3.8 | 4.1 | 4.4 |
| Croatia | cases | 374 | 2020-06-29 | 3.2 | 3.5 | 3.9 |
| Burkina_Faso | cases | 56 | 2020-06-29 | 1.9 | 2.5 | 3.2 |
| Montenegro | cases | 119 | 2020-06-29 | 2.1 | 2.5 | 3.0 |
| Luxembourg | cases | 122 | 2020-06-29 | 2.0 | 2.4 | 2.8 |
| Palestine | cases | 1,226 | 2020-06-29 | 2.2 | 2.3 | 2.4 |
| Angola | cases | 91 | 2020-06-29 | 1.7 | 2.1 | 2.5 |
| Uruguay | cases | 53 | 2020-06-29 | 1.6 | 2.0 | 2.5 |
Below we plot results for each country/province in a list. We filter out weeks where the upper end of confidence interval for \(R_{t,m}\) exceeds five.
Below we plot estimates for all the provinces. Note that [1] recommends only to start using estimates after at least 12 cases have occurred. On that basis we do not have estimates for R based on deaths for a number of provinces.
Below we plot all the countries:
Below we plot estimates of \(R_{t,m}\) based on cases versus those based on deaths for countries where these estimates range between 0.5 and 2.
Below we plot where there were more than 5 deaths and cases in the last week. It appears that, generally, \(R_{t,m}^{cases}\) and \(R_{t,m}^{deaths}\) are consistent.
Below we plot where there were more than 25 deaths and cases in the last week. It appears that, generally, \(R_{t,m}^{cases}\) and \(R_{t,m}^{deaths}\) are consistent.
Below we plot weeks of estimates of \(R_{t,m}\) versus the average mobility index that was constructed. There is a clear link between mobility and the reproductive number. As it decreases \(R_{t,m}\) reduces:
TO DO
From the basic plots it’s clear that South Africa and a few other countries appear to be on a different “slope” than European countries show. These include Brazil, Peru, Chile and India.
The above shows estimates for reproductive number using cases and deaths (\(R_{t,m}^{cases}\) and \(R_{t,m}^{deaths}\)) for each country \(m\) over time \(t\) in weeks.
From the current estimates it appears that, at present, in general South African provincial estimates based on cases and deaths correspond. An exception to this may be Eastern Cape with estimates based on deaths exceeding those based on cases.
It is also clear that estimates based on cases appear to precede the movements based on deaths over time. This can be expected as, all things being equal, infections and associated cases should be a precursor to deaths.
South Africa does not compare well, and appears to have one of the highest \(R_{t,m}^{deaths}\) in the world. This is higher than the \(R_{t,m}^{cases}\) which may be indicative of lower testing and/or backlogs with regard to testing.
On a scatter plot of countries most appear to have \(R_{t,m}^{cases}\) correlated well with \(R_{t,m}^{deaths}\). Chile and Afghanistan appear to be outliers with \(R_{t,m}^{deaths}\) higher than \(R_{t,m}^{cases}\) indicating perhaps limited testing, or alternatively epidemics that are slowing (given the leading nature of cases vs. deaths).
Overall it’s clear that having multiple measures of \(R_{t,m}\) appears useful and monitoring it’s value is something useful.
We also show that mobility is clearly correlated with \(R_{t,m}^{cases}\) in the countries shown.
Limitation of this method to estimate \(R_{t,m}\) are noted in [1]
Despite these limitation we believe the ease of calculation of this method and the ability to use multiple sources makes it useful as a monitoring tool.
Detailed output for all countries are saved to a comma-separated value file. The file can be found here.
[1] A. Cori, N. M. Ferguson, C. Fraser, and S. Cauchemez, “A new framework and software to estimate time-varying reproduction numbers during epidemics,” American Journal of Epidemiology, vol. 178, no. 9, pp. 1505–1512, Sep. 2013, doi: 10.1093/aje/kwt133. [Online]. Available: https://doi.org/10.1093/aje/kwt133
[2] A. Cori, EpiEstim: A package to estimate time varying reproduction numbers from epidemic curves. 2013 [Online]. Available: https://CRAN.R-project.org/package=EpiEstim
[3] V. Marivate et al., “Coronavirus disease (COVID-19) case data - South Africa.” Zenodo, 2020 [Online]. Available: https://zenodo.org/record/3888499
[4] European Centre for Disease Prevention and Control, “Data on the geographic distribution of COVID-19 cases worldwide.” European Union, 2020 [Online]. Available: https://www.ecdc.europa.eu/en/publications-data/download-todays-data-geographic-distribution-covid-19-cases-worldwide
[5] R. Minder, “Counting bodies and pointing fingers as Spain tallies coronavirus dead,” New York Times [Online]. Available: https://www.nytimes.com/2020/04/16/world/europe/coronoavirus-spain-death-toll.html
[6] Google LLC, “Google COVID-19 community mobility reports.” 2020 [Online]. Available: https://www.google.com/covid19/mobility/
[7] P. Nouvellet et al., “Report 26: Reduction in mobility and COVID-19 transmission,” Imperial College London, 2020 [Online]. Available: https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-26-mobility-transmission/
[8] N. Ferguson et al., “Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand,” Imperial College London, 2020 [Online]. Available: https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-9-impact-of-npis-on-covid-19/